## Efficient numerical methods in non-uniform sampling theory (1995)

### Cached

### Download Links

Citations: | 78 - 9 self |

### BibTeX

@MISC{Feichtinger95efficientnumerical,

author = {Hans G. Feichtinger and Karlheinz Gröchenig and Thomas Strohmer},

title = { Efficient numerical methods in non-uniform sampling theory},

year = {1995}

}

### OpenURL

### Abstract

We present a new “second generation” reconstruction algorithm for irregular sampling, i.e. for the problem of recovering a band-limited function from its non-uniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems. The choice of ”adaptive weights” can be seen as a simple but very efficient method of preconditioning. Further substantial acceleration is achieved by utilizing the Toeplitztype structure of the system matrix. This new algorithm can handle problems of much larger dimension and condition number than have been accessible so far. Furthermore, if some gaps between samples are large, then the algorithm can still be used as a very efficient extrapolation method across the gaps.

### Citations

635 |
Matrix Iterative Analysis
- Varga
- 2000
(Show Context)
Citation Context ...1 p n = p for p 2 PM and kp 0 p n k 2sfl n kpk 2 (13) where fl = maxfj1 0 Aj; j1 0 Bjg ! 1. Since Sp depends only on the samples p(t i ), this is indeed a reconstruction from the samples only. Proof. =-=[11, 23, 39, 42, 43]-=- Since hSp; pi = P r i=1 jp(t i )j 2 by (10) and (11), S is a positive operator on PM and thus invertible. By (8) one obtains (1 0 B)kpk 2 2sh(Id 0 S)p; pis(1 0 A)kpk 2 2 (14) and consequently kp 0 p ... |

335 |
Iterative Solution of Large Linear Systems
- Young
- 1971
(Show Context)
Citation Context ...1 p n = p for p 2 PM and kp 0 p n k 2sfl n kpk 2 (13) where fl = maxfj1 0 Aj; j1 0 Bjg ! 1. Since Sp depends only on the samples p(t i ), this is indeed a reconstruction from the samples only. Proof. =-=[11, 23, 39, 42, 43]-=- Since hSp; pi = P r i=1 jp(t i )j 2 by (10) and (11), S is a positive operator on PM and thus invertible. By (8) one obtains (1 0 B)kpk 2 2sh(Id 0 S)p; pis(1 0 A)kpk 2 2 (14) and consequently kp 0 p ... |

325 |
Toeplitz Forms and their Applications
- Grenander, Szego
- 1958
(Show Context)
Citation Context ...e computed at any point t. In particular, p can be evaluated on any grid fn=N : n = 0; : : : N01g by FFT. All our plots of reconstructions have been obtained in this way. 3. By Caratheodory's theorem =-=[19] eve-=-ry positive definite Toeplitz matrix T can be written as T lk = r X j=1 w j e 2��i(l0k)t j for some uniquely determined sequence w j ? 0 and t j 2 [0; 1). From this point of view the use of the we... |

291 |
Circulant Matrices
- Davis
- 1979
(Show Context)
Citation Context ...umn of C. Applying the discrete Fourier transform to this equation it becomessca =sb. Thereforesa is given by a component-by-component division, and a is recovered fromsa by inverse Fourier transform =-=[35, 9]-=-. Multiplication of a vector a by a Toeplitz matrix T can also be carried out quickly by means of an appropriate FFT. The n2n matrix T is extended to a circulant ~ T of size (2n01)2(2n0 1), and the ve... |

253 |
A Class of Nonharmonic Fourier Series
- Dufi, Schaeffer
- 1952
(Show Context)
Citation Context ...thm. Let DM denote the Dirichlet kernel DM (t) = M X k=0M e 2��ikt = sin(M + 1 2 )2��t sin ��t (9) and observe that p(t) = Z 1 0 p(u)DM (t 0 u) du = hp; DM (: 0 t)i : (10) Define the frame=-= operator S [11, 43]-=- by Sp(t) = r X i=1 p(t i )DM (t 0 t i ) : (11) Following Duffin and Schaeffer [11] we obtain the following simple iterative reconstruction algorithm. Lemma 2 Fix M 2 IN and suppose rs2M + 1 ands! 1 B... |

196 |
Signal Analysis
- Papoulis
- 1977
(Show Context)
Citation Context ...ctions. In the ideal case of equally spaced samples the reconstruction is routine and can be carried out explicitly by one of the many variations of the ShannonWhittaker -Kotel'nikov sampling theorem =-=[6, 25, 32]-=-. However, in many applications, for instance in astronomy, seismology, tomography and physics, one is forced to sample signals at nonuniformly 1 The second named author was partially supported by NSF... |

182 |
An Introduction to Non-Harmonic Fourier Series
- Young
- 1980
(Show Context)
Citation Context ...thm. Let DM denote the Dirichlet kernel DM (t) = M X k=0M e 2��ikt = sin(M + 1 2 )2��t sin ��t (9) and observe that p(t) = Z 1 0 p(u)DM (t 0 u) du = hp; DM (: 0 t)i : (10) Define the frame=-= operator S [11, 43]-=- by Sp(t) = r X i=1 p(t i )DM (t 0 t i ) : (11) Following Duffin and Schaeffer [11] we obtain the following simple iterative reconstruction algorithm. Lemma 2 Fix M 2 IN and suppose rs2M + 1 ands! 1 B... |

177 | Applied Iterative Methods - Hageman, Young - 1981 |

93 |
An optimal circulant preconditioner for Toeplitz systems
- Chan
- 1988
(Show Context)
Citation Context ...e preconditioned conjugate gradient method (referred to as PCG) will converge much faster than the original CG method. Various efficient preconditioners have been developed in the last ten years, see =-=[7, 28, 26, 8]-=- for detailed discussions of preconditioners. In the present paper we use the optimal circulant Frobenius norm approximation C F introduced by T. Chan [8]. We have mentioned in Section 4 that the use ... |

84 |
Asymptotically fast solution of Toeplitz and related systems of linear equations, Linear Algebra and its Applications 34
- Bitmead, Anderson
- 1980
(Show Context)
Citation Context ...�ikt j k = 0; 1; : : : ; 2M : (25) (c) The matrix T of equation (19) is a Toeplitz matrix for any spectrum of the form [M 1 ; M 2 ]. (d) There is a large repertory of Toeplitz solvers at our disposa=-=l [1, 35, 36, 3, 10]-=- which can solve (23) with only O(M 2 ); O(M log 2 M) or even O(M log M) operations. Depending on the size of M and the required precision one may choose either (I) direct methods for the complete inv... |

63 |
Stability of methods for solving Toeplitz systems of equations
- Bunch
- 1985
(Show Context)
Citation Context ...ut special structure. A number of so-called "superfast" direct inversion methods [1, 10] have been created in the last ten years. However the stability of these fast direct solvers is still =-=a problem [5]-=-. Furthermore, since in many applications a solution is required only with a certain accuracy, but not the exact solution, we prefer to use iterative methods. In [35] Strang proposed an iterative meth... |

63 |
A proposal for Toeplitz matrix calculations
- Strang
- 1986
(Show Context)
Citation Context ...�ikt j k = 0; 1; : : : ; 2M : (25) (c) The matrix T of equation (19) is a Toeplitz matrix for any spectrum of the form [M 1 ; M 2 ]. (d) There is a large repertory of Toeplitz solvers at our disposa=-=l [1, 35, 36, 3, 10]-=- which can solve (23) with only O(M 2 ); O(M log 2 M) or even O(M log M) operations. Depending on the size of M and the required precision one may choose either (I) direct methods for the complete inv... |

55 |
Theory and practice of irregular sampling
- Feichtinger, Grchenig
- 1994
(Show Context)
Citation Context ...nometric interpolation, band-limited functions, irregular sampling, conjugate gradient, Toeplitz matrix, preconditioner. spaced points. This problem has received much attention in the past years, see =-=[2, 30, 15, 14, 17]-=- for history and references. But despite an abundance of work on the irregular sampling problem --- [29] lists about 300 references --- its numerical and algorithmic aspects have been neglected so far... |

54 | Superfast solution of real positive definite Toeplitz systems
- Ammar, Gragg
- 1988
(Show Context)
Citation Context ...�ikt j k = 0; 1; : : : ; 2M : (25) (c) The matrix T of equation (19) is a Toeplitz matrix for any spectrum of the form [M 1 ; M 2 ]. (d) There is a large repertory of Toeplitz solvers at our disposa=-=l [1, 35, 36, 3, 10]-=- which can solve (23) with only O(M 2 ); O(M log 2 M) or even O(M log M) operations. Depending on the size of M and the required precision one may choose either (I) direct methods for the complete inv... |

44 |
Reconstruction algorithms in irregular sampling
- Gröchenig
- 1992
(Show Context)
Citation Context ...e of work on the irregular sampling problem --- [29] lists about 300 references --- its numerical and algorithmic aspects have been neglected so far. Simple iterative algorithms have been proposed in =-=[2, 13, 17, 20, 22, 31, 34, 40, 41]-=-. These algorithms seem to work decently for well-conditioned problems and for small data sets, but become slow and expensive for more complicated and more realistic problems. A comparison of the perf... |

43 | Toeplitz equations by conjugate gradients with circulant preconditioner - Chan, Strang - 1989 |

38 |
The inversion of finite Toeplitz matrices and their continual analogues
- Gohberg, Semencul
- 1972
(Show Context)
Citation Context ...h. Remark: If many signals of the same bandwidth have to be reconstructed from the same sampling geometry, it is useful to establish the inverse of the Toeplitz matrix in the Gohberg-Semencul formula =-=[18]-=- once, which can be easily done by a slight modification of ACT. The reconstruction of the signals can then be done considerably faster [27, 37, 38]. 8 Numerical Results In this section we will discus... |

34 |
Five short stories about the cardinal series
- Higgins
- 1985
(Show Context)
Citation Context ...ctions. In the ideal case of equally spaced samples the reconstruction is routine and can be carried out explicitly by one of the many variations of the ShannonWhittaker -Kotel'nikov sampling theorem =-=[6, 25, 32]-=-. However, in many applications, for instance in astronomy, seismology, tomography and physics, one is forced to sample signals at nonuniformly 1 The second named author was partially supported by NSF... |

31 |
Iterative reconstruction of band-limited images from nonuniformly spaced samples
- Sauer, Allebach
- 1987
(Show Context)
Citation Context ...e of work on the irregular sampling problem --- [29] lists about 300 references --- its numerical and algorithmic aspects have been neglected so far. Simple iterative algorithms have been proposed in =-=[2, 13, 17, 20, 22, 31, 34, 40, 41]-=-. These algorithms seem to work decently for well-conditioned problems and for small data sets, but become slow and expensive for more complicated and more realistic problems. A comparison of the perf... |

29 |
Iterative reconstruction of multivariate band-limited functions from irregular sampling values
- Feichtinger, Grochenig
- 1992
(Show Context)
Citation Context ...nometric interpolation, band-limited functions, irregular sampling, conjugate gradient, Toeplitz matrix, preconditioner. spaced points. This problem has received much attention in the past years, see =-=[2, 30, 15, 14, 17]-=- for history and references. But despite an abundance of work on the irregular sampling problem --- [29] lists about 300 references --- its numerical and algorithmic aspects have been neglected so far... |

29 |
Acceleration of the frame algorithm
- Gröchenig
- 1993
(Show Context)
Citation Context ... at most kx 0 x n kAs2 /p 3 0 p p 3 + p !n kx 0 x 0 kA (37) where kxkA = hx; Axi 1=2 is the A-norm of x. Consequently we could accelerate each of the simple iterations of Lemma 2, 3, and (4) and (29) =-=[21]-=-. The immediate advantages are: 1. Improvement of the convergence by an order of magnitude. 2. If the error is measured with respect to the operator used in the iteration (S; Sw ; T , or Tw ), then th... |

29 |
A discrete theory of irregular sampling
- Gröchenig
- 1993
(Show Context)
Citation Context ...e of work on the irregular sampling problem --- [29] lists about 300 references --- its numerical and algorithmic aspects have been neglected so far. Simple iterative algorithms have been proposed in =-=[2, 13, 17, 20, 22, 31, 34, 40, 41]-=-. These algorithms seem to work decently for well-conditioned problems and for small data sets, but become slow and expensive for more complicated and more realistic problems. A comparison of the perf... |

28 |
Discrete least squares approximation by trigonometric polynomials
- Reichel, Ammar, et al.
- 1991
(Show Context)
Citation Context ...e which interpolates the given data (t i ; y i ); i = 1; : : : ; r; in the sense that p(t i ) = y i . There are several explicit interpolation procedures known, see [44] and also efficient algorithms =-=[4, 33]-=-. However, numerically these methods seem to be fairly unstable. In practice an upper bound for the degree of p is known as a consequence of the band-limitedness and one avoids the bad conditioning by... |

27 |
Recovery of signals from nonuniform samples using iterative methods
- Marvasti, Analoui, et al.
- 1991
(Show Context)
Citation Context |

26 |
The sampling theorems and linear prediction in signal analysis
- Butzer, Splettster, et al.
- 1988
(Show Context)
Citation Context ...ctions. In the ideal case of equally spaced samples the reconstruction is routine and can be carried out explicitly by one of the many variations of the ShannonWhittaker -Kotel'nikov sampling theorem =-=[6, 25, 32]-=-. However, in many applications, for instance in astronomy, seismology, tomography and physics, one is forced to sample signals at nonuniformly 1 The second named author was partially supported by NSF... |

25 |
Iterative Losung groer schwachbesetzter Gleichungssysteme
- Hackbusch
- 1991
(Show Context)
Citation Context ...1 p n = p for p 2 PM and kp 0 p n k 2sfl n kpk 2 (13) where fl = maxfj1 0 Aj; j1 0 Bjg ! 1. Since Sp depends only on the samples p(t i ), this is indeed a reconstruction from the samples only. Proof. =-=[11, 23, 39, 42, 43]-=- Since hSp; pi = P r i=1 jp(t i )j 2 by (10) and (11), S is a positive operator on PM and thus invertible. By (8) one obtains (1 0 B)kpk 2 2sh(Id 0 S)p; pis(1 0 A)kpk 2 2 (14) and consequently kp 0 p ... |

23 |
Irregular sampling theorems and series expansions of band-limited functions
- Feichtinger, Grochenig
- 1992
(Show Context)
Citation Context ...nometric interpolation, band-limited functions, irregular sampling, conjugate gradient, Toeplitz matrix, preconditioner. spaced points. This problem has received much attention in the past years, see =-=[2, 30, 15, 14, 17]-=- for history and references. But despite an abundance of work on the irregular sampling problem --- [29] lists about 300 references --- its numerical and algorithmic aspects have been neglected so far... |

23 |
On discrete band-limited signal extrapolation
- Strohmer
- 1995
(Show Context)
Citation Context ...erse of the Toeplitz matrix in the Gohberg-Semencul formula [18] once, which can be easily done by a slight modification of ACT. The reconstruction of the signals can then be done considerably faster =-=[27, 37, 38]-=-. 8 Numerical Results In this section we will discuss some numerical results of the proposed algorithm. We will demonstrate the efficiency of the ACT algorithm and illustrate how appropriate precondit... |

19 |
Circulant preconditioned toeplitz least squares iterations, preprint
- Chan, Nagy, et al.
(Show Context)
Citation Context ...e preconditioned conjugate gradient method (referred to as PCG) will converge much faster than the original CG method. Various efficient preconditioners have been developed in the last ten years, see =-=[7, 28, 26, 8]-=- for detailed discussions of preconditioners. In the present paper we use the optimal circulant Frobenius norm approximation C F introduced by T. Chan [8]. We have mentioned in Section 4 that the use ... |

15 | Error analysis in regular and irregular sampling theory - Feichtinger, Gröchenig - 1993 |

15 |
Efficient methods for digital signal and image reconstruction from nonuniform samples
- STROHMER
- 1993
(Show Context)
Citation Context ...with the conjugate gradient acceleration provides an efficient method for the reconstruction of bandlimited signals from non-uniform samples [21]. A detailed discussion of this method can be found in =-=[37]-=-. 6 Superfast Reconstruction from Irregular Samples By combining the reformulation of the original problem as a Toeplitz system with the adaptive weights method and with the conjugate gradient acceler... |

15 | Advanced Topics in Shannon Sampling and Interpolation Theory - Marks - 1992 |

14 |
Recovery of bandlimited signals from unequally spaced samples
- Wiley
(Show Context)
Citation Context |

13 |
A Unified Approach to Zero-Crossings and Nonuniform Sampling
- Marvasti
- 1987
(Show Context)
Citation Context |

11 |
Irregular sampling and the theory of frames
- Benedetto, Heller
(Show Context)
Citation Context |

11 |
Iterative algorithms in irregular sampling a first comparison of methods
- Cenker, Feichtinger, et al.
(Show Context)
Citation Context ...small data sets, but become slow and expensive for more complicated and more realistic problems. A comparison of the performance of the "first generation" of reconstruction algorithms can be=-= found in [12, 17]. In the p-=-resent paper we introduce a new "superfast" algorithm for the reconstruction of band-limited signals from irregular samples. This algorithm is iterative and cuts the number of iterations to ... |

11 |
Iterative and one-step reconstruction from nonuniform samples by convex projections
- Yen, Stark
- 1990
(Show Context)
Citation Context |

8 |
Design and analysis of Toeplitz preconditioners
- KU, KUO
- 1992
(Show Context)
Citation Context ...e preconditioned conjugate gradient method (referred to as PCG) will converge much faster than the original CG method. Various efficient preconditioners have been developed in the last ten years, see =-=[7, 28, 26, 8]-=- for detailed discussions of preconditioners. In the present paper we use the optimal circulant Frobenius norm approximation C F introduced by T. Chan [8]. We have mentioned in Section 4 that the use ... |

8 | Iterative Lösung großer schwachbesetzter Gleichungssysteme. Teubner Studienbücher - Hackbusch - 1993 |

6 |
Some Aspects of Circulant Preconditioners
- Huckle
- 1992
(Show Context)
Citation Context |

5 |
Solution of Vandermonde systems of equations
- Björk, Pereyra
(Show Context)
Citation Context ...e which interpolates the given data (t i ; y i ); i = 1; : : : ; r; in the sense that p(t i ) = y i . There are several explicit interpolation procedures known, see [44] and also efficient algorithms =-=[4, 33]-=-. However, numerically these methods seem to be fairly unstable. In practice an upper bound for the degree of p is known as a consequence of the band-limitedness and one avoids the bad conditioning by... |

4 |
A new algorithm for solving Toeplitz system of equations
- Hoog
- 1987
(Show Context)
Citation Context |

4 |
Fast iterative and non-iterative reconstruction methods in irregular sampling
- Feichtinger, Cenker, et al.
- 1991
(Show Context)
Citation Context |

2 |
Mailing address
- Press
- 1910
(Show Context)
Citation Context ...ric polynomial of appropriate degree which interpolates the given data (t i ; y i ); i = 1; : : : ; r; in the sense that p(t i ) = y i . There are several explicit interpolation procedures known, see =-=[44]-=- and also efficient algorithms [4, 33]. However, numerically these methods seem to be fairly unstable. In practice an upper bound for the degree of p is known as a consequence of the band-limitedness ... |

1 |
An efficient algorithm for a large Toeplitz system of equations
- Jain
- 1979
(Show Context)
Citation Context ...erse of the Toeplitz matrix in the Gohberg-Semencul formula [18] once, which can be easily done by a slight modification of ACT. The reconstruction of the signals can then be done considerably faster =-=[27, 37, 38]-=-. 8 Numerical Results In this section we will discuss some numerical results of the proposed algorithm. We will demonstrate the efficiency of the ACT algorithm and illustrate how appropriate precondit... |

1 | Applied Iterative Methods. Academic Press Numerische Mathematik Electronic Edition -- page numbers may differ from the printed version page 440 - Hageman, Young - 1981 |

1 | Trigonometric Series, Vol. II. Cambridge Univ. Press This article was processed by the author using the LÉ T E X style file pljour1 from Springer-Verlag - Zygmund - 1959 |